Th\'eorie $L^p$ pour l'\'equation de Cauchy-Riemann

TL;DR

This paper develops a comprehensive $L^p$-theory for the Cauchy-Riemann operator on complex manifolds, including local, Andreotti-Grauert, and Serre duality aspects, advancing the understanding of solvability in $L^p$ spaces.

Contribution

It introduces a systematic $L^p$ framework for the Cauchy-Riemann operator, extending classical theories to $L^p$-spaces with new duality and solvability results.

Findings

Established $L^p_{loc}$-theory for the Cauchy-Riemann operator.

Developed an $L^p$ Andreotti-Grauert theory.

Analyzed Serre duality and applications to solvability in $L^p$-spaces.

Abstract

In this paper we propose a systematic study of the Cauchy-Riemann operator in the $L^p$-setting in complex manifolds. We first consider $L^p_{loc}$-theory and then we develop an $L^p$ Andreotti-Grauert theory. Finally we consider Serre duality and its applications to the solvability of the Cauchy-Riemann equation with exact support in $L^p$-spaces.